International Journal of Multiphase Flow 35 (2009) 485–497

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International Journal of Multiphase Flow

journal homepage: www.elsevier .com/ locate / i jmulflow

Liquid/liquid dispersion in a chaotic advection flow

Charbel Habchi a,b, Thierry Lemenand a, Dominique Della Valle a, Hassan Peerhossaini a,*

a Thermofluids, Complex Flows and Energy Research Group, Laboratoire de Thermocinétique de Nantes, CNRS UMR 6607, École Polytechnique – Université de Nantes,rue Christian Pauc, B.P. 50609, 44306 Nantes, Franceb Agence de l’Environnement et de la Maîtrise de l’Énergie (ADEME), 20 avenue du Grésillé, B.P. 90406, 49004 Angers, France

a r t i c l e i n f o a b s t r a c t

Article history:Received 12 September 2008Received in revised form 10 February 2009Accepted 22 February 2009Available online 12 March 2009

Keywords:Chaotic advectionLiquid/liquid dispersionDrop size distributionMultifunctional heat exchanger

0301-9322/$ - see front matter � 2009 Elsevier Ltd. Adoi:10.1016/j.ijmultiphaseflow.2009.02.019

* Corresponding author. Tel.: +33 2 40 68 31 24; faE-mail address: hassan.peerhossaini@univ-nantes.

Mixing by chaotic advection in a twisted-pipe flow is used here to investigate the efficiency of this flow inthe liquid/liquid dispersion process. This study focuses on water/oil dispersions produced by continuouswater injection into a main oil flow, for small Dean numbers. The drop sizes obtained with the chaotic-advection twisted-pipe flow are compared with those in a straight pipe and a helically coiled flow for thesame conditions. It is found that the resulting dispersions are finer and more mono-dispersed in the cha-otic advection flow. These results are compared with the theoretical maximum diameter dmax determinedby the Grace theory in which the viscous stress controls the breakup phenomena. For this purpose, thekinematic field is computed from the theoretical formulae for Dean flow. The strain rate fields in the pipecross-section are then analytically computed and used to predict the maximum drop diameter. The the-oretical values are identical for the three configurations (straight, helically coiled, and twisted pipe) up toa critical Dean number, where the secondary flow becomes significant. Beyond this value, the shear stressis enhanced in the twisted-pipe flow compared with the straight-pipe flow, and the predicted drop diam-eters are smaller. An interpretation of the higher dispersive performance of the chaotic flow is providedby the Lagrangian trajectories of the particles.

� 2009 Elsevier Ltd. All rights reserved.

1. Introduction

The advantages offered by a chaotic-advection twisted pipe-flow used as a mixer and/or heat exchanger have been establishedin previous studies (Aref, 1984; Jones et al., 1989; Acharya et al.,1992; Peerhossaini et al., 1993; Castelain et al., 1997; Mokraniet al., 1998a,b; Le Guer et al., 2001; Lemenand and Peerhossaini,2002) when compared with a straight pipe or helically coiled pipe.The geometric perturbation of the twisted-pipe configuration gen-erates three-dimensional chaotic trajectories in the secondaryDean flow induced by curvature effects. The Dean number charac-terizes the ratio between the viscous forces and the centrifugalforces, and is defined as

De ¼W2a3

Rm2 ð1Þ

where a is the tube inner radius and R the bend curvature radius.Chaotic advection in such geometries produces efficient macro-

mixing and heat transfer in the laminar regime. The heat transfer isenhanced in a particular range of the Dean number [60–1000], with-out significant increase in the pressure drop (Mokrani et al., 1998a,b).

This flow finds applications in the pharmaceutical industry orfood industry to process highly viscous fluids or fluids with

ll rights reserved.

x: +33 2 40 68 31 41.fr (H. Peerhossaini).

stress-sensitive long molecular chains. The purpose of this studyis to investigate the capacity of chaotic advection to generateliquid/liquid dispersion.

An experimental study was undertaken in order to observe theeffect of chaotic advection on the dispersion of water in a laminaroil flow, and also to obtain experimental data for validation of theresults of a theoretical approach to the water–oil dispersion in thisflow. The theoretical approach is based on the determination ofdrop equilibrium under the joint action of viscous stresses gener-ated by chaotic advection flow and interface tension.

In a curved pipe, the centrifugal force induces a secondary flowin the form of counter-rotating cells called Dean roll-cells that aresuperimposed on the axial flow and play the role of internal agita-tors of the flow. Analytical solutions for Dean flow have been pro-posed by Jones et al. (1989) and Le Guer and Peerhossaini (1991).

In the present work, the theoretical determination of the equi-librium drop size is based on Taylor’s analysis (Taylor, 1953) asextended by Grace (1982). From this it is possible to computethe Eulerian distribution of the strain rates in the pipe cross-sec-tion, and to propose a model for the maximum shear and elonga-tion rates introduced in Grace’s work (1982) to calculate thetheoretical maximum diameters with no fitting constant. The addi-tional forces introduced by Dean flow above a certain Dean numbergenerate stresses due to stretching and folding that are over andabove the basic strain rates in the straight pipe, and these stressesenhance the breakup of water drops injected into the main oil flow.

Fig. 2. Twisted pipe test section.

486 C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497

From this point of view, there is no theoretical difference betweenthe helically coiled and twisted-pipe chaotic advection configura-tions. The drop diameters predicted by the theoretical approachare then compared with the experimental values for the three con-figurations (straight, helically coiled, and twisted-pipe flows).

A Lagrangian analysis of the flow was undertaken to explain thefiner drop size observed experimentally in the chaotic advectionflow. A comparison of the trajectories and of the Poincaré sectionof the fluid particles after 25 bends, for helically coiled and chaoticadvection flows, qualitatively confirms that the latter configurationprovides better macro-mixing. This feature is clear in the residencetime distribution (RTD) for the two configurations and helps toexplain the better dispersive performance of the chaotic advectionflow. An illustration for De ¼ 100 shows the improved homogeniz-ing properties of the chaotic advection configuration.

This paper is organized as follows. In Section 2 the experimentalsetup and experimental results for the water–oil dispersion arepresented. Section 3 is devoted to the theoretical prediction ofdroplet diameters by an Eulerian approach. The mechanical historyof droplets in the flow determined by Lagrangian approach is givenin Section 4, as is a presentation of the resident time distribution offluid particles. Conclusions are drawn in Section 5.

Table 1Test section dimensions.

Diameter of circular duct 8 mmBend curvature radius 44 mmCurvature angle in bend plane p

2 radNumber of bends 25Total curved length 1.8 mStraight length between bends 0.2 mTotal length 2 m

2. Experimental setup and methods

2.1. Experimental setup

As shown in Fig. 1, the test section is composed of a successionof 90� bends with a given curvature radius. It can be arranged inboth the helically coiled pipe and the twisted-pipe configurationswhere each bend is rotated in the orthogonal plane with respectto the preceding one, as seen in Fig. 2. Both geometries have 25bends of 8 mm inside diameter D, the same unfolded lengthðL ¼ 2 mÞ and the same bend curvature radius ðR ¼ 44 mmÞ. Thestraight pipe has also an inside diameter of D ¼ 8 mm andunfolded length L ¼ 2 m. The dimensions of the test section aregiven in Table 1.

A schematic diagram of the hydraulic loop used here is shownin Fig. 3a. It has four main elements:

– oil admission circuit;– water injection system;– test section;– flow visualization device at the test section exit.

The tank contains 100 L of a vegetable oil, allowing experimen-tation for 1 h at a Reynolds number of Re ¼ 50 and 5 min at

Centrifugalforce F

Dean roll-cells

(a)

Fig. 1. Generation of spatially chaotic particle paths in laminar, steady, and thre

Re ¼ 600. The flow rates are measured by rotameters with 5% accu-racy (Sart Von Rohr SASTM for water and Krohne DusburgTM for oil).

The oil is pumped by a centrifugal pump, while the water issupplied by a constant-level feed tank connected to the watersupply network. The flow rate is monitored by a back-pressurevalve. Water is injected at the twisted pipe inlet by an injectionneedle of inside diameter 2 mm, designed so that the velocity ratio(injection/main flow) does not exceed 1.5 for the most extremeconditions (water volume fraction of 10%). The current volumefractions are much lower, so that perturbations at the injectionmight not induce an additional breakup of the water droplets.

2.2. Drop size measurement

The flow visualization system (Fig. 3b) is a rectangularPlexiglasTM window positioned on top of a parallelepipedic box,

F

F

Chaotictrajectories

Flow

(b)

e-dimensional flow: (a) regular Dean flow; and (b) twisted pipe Dean flow.

Flowmeters

Test section

Setting tank

Visualization box

Oil tank

Water tank

Centrifugal pump

25 cm11 cm

2.3 cm

Pipe outlet

Conic shape diffusersConic shape nozzle

5 cm

2.3 cm

1.3 cm

weivtnorFweivpoT

(a)

(b)

Fig. 3. (a) Schematic diagram of experimental setup. (b) Flow visualization system.

C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497 487

with conic/rectangular connections at each end. At the entrance,directly at the exit of the test section, the diffuser, is designed tominimize flow disturbances by maintaining (as much as possible)the shear stresses at the same level as in the test section, and, witha 7� angle, to avoid recirculations at low flow rates and thus to pre-vent drop coalescence at the pipe exit. Moreover, the box depth issmall enough (depth is 13 mm, equals to 1.625 D) to prevent dropsoverlapping. In fact overlapping did not occur for the range of flowrates in this experiment, but may occur for higher flow rates.

Pictures of the emulsion are taken with a high-frequency digitalCanonTM camera placed vertically above the visualization windowwith its optical axis perpendicular to the window’s plane. Theemulsion flow in the visualization box is lit from below by anintense diffuse white light. For given operating conditions, asequence of independent images is selected and recorded; thissequence, an example of which is shown in Fig. 4, constitutes ourstatistical sample of the drops. Drop diameters are measured fromthe recorded images using IMAQ Vision Builder 6 software. At theend of the analysis, a table of diameters of at least 400 drops foreach run is obtained. The experimental size distributions are fittedwith a log-normal law. By taking 99% of the cumulative volume

Fig. 4. Image of droplets taken by fast digital camera (chao

curve, a representative value for the maximum diameter can bedetermined. The standard deviation factor is also of interest forthe quality of the emulsification process in further applications. Ifk is the log-standard deviation, the linear deviation b can bededuced from

b ¼ sinhðkÞ ð2Þ

Measurements were carried out for the three configurations:straight pipes, helically coiled pipes, and twisted pipes. By thelog-normal law, the maximum drop diameter is found by reading99% from the frequency of the cumulative diameter values on thefitted log-normal curve, as shown in Fig. 5 for a straight tube.

2.3. Working fluids

The continuous phase is commercial food-grade vegetable oil,and tap water is injected as the dispersed phase. The kinematic vis-cosity of the oil was measured using a MettlerTM RM180 rheometer.To measure the surface tension of the working fluid we used theWilhelmy method: a metallic blade suspended from a balance bya stem is plunged into the liquid; the balance measures the vertical

tic advection configuration – oil flow rate Q = 50 l h�1).

500 600 700 800 900 1000 1100

0.0

0.2

0.4

0.6

0.8

1.0

Fre

qu

ency

Diameters (µm)

Experimental

Log normal law

experimental dmax

99%

Fig. 5. Experimental droplet diameter distribution fitted by log-normal curve (straight pipe with oil flow rate Q = 80 l h�1).

Table 3

488 C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497

force F exerted on the stem by withdrawing the blade from theliquid. The surface tension r is calculated by:

F ¼ rf cos h ð3Þ

where f is the wet perimeter. The platinum blade is previouslypassed through a flame to obtain perfect adhesion to the liquid sur-face. The ideal contact angle is h = 0�, for which the term cosh in Eq.(2) tends to 1. Therefore the value of the surface tension r can bededuced from the geometric characteristics of the blade.

As the oil viscosity lc is very sensitive to temperature and hassignificant consequences on the dispersion through the viscousstress, the oil temperature T was controlled by a Chromel/Alumel(type K) thermocouple in the oil admission circuit. For modellingpurposes, oil viscosity was measured with an AR1000 TA InstrTM

rheometer and fitted by Arrhenius’ law:

lc ¼ lc0exp

ERT� E

RT0

� �ð4Þ

where E = 29 kJ, R = 8.314 J K�1 mol�1 and lc0is the oil viscosity for

a given temperature T0. The physical properties of the two fluids aregiven in Table 2.

2.4. Reproducibility

Runs were repeated three times on different days to check theeffect of a new operator and a new trial on the measured character-istics of the final emulsion in the test section. Nominal operatingconditions at Dean number De ¼ 70; the resulting size distribu-tions of the three runs were compared, leading to a maximum errorof ±6% in the maximum drop diameter.

Table 2Characteristics of working fluids used in experiments.

Interfacial tension 0.0192 N m�1

Oil density 910 kg m�3

Water density 1000 kg m�3

Water dynamic viscosity 0.001 Pa sOil dynamic viscosity at 25 �C 0.052 Pa sViscosity ratio 0.0192

2.5. Experimental results for drop diameters

Drop size distributions were measured for the three configura-tions: straight, helically coiled and twisted pipes. A summary of theresults is given in Table 3.

The maximum drop diameters are plotted versus the oil flowrate in Fig. 6. As expected, the drop diameters decrease withincreasing flow rate, since the viscous stresses increase with yieldvelocity and velocity gradients. The water (dispersed phase) flowrate is not taken into account as an operating parameter unless itincreases the global (two-phase) flow rate. This approximation isreasonable as long as we work with the lowest injection rate(about 5% volume fraction) to minimize flow disturbance at theinjection point and to prevent drop coalescence: at this level, thedispersed phase volume fraction does not influence the dropbreakup. The drop diameters obtained in the chaotic flow are smal-ler than those in the helically coiled flow, the latter being smallerthan that in the straight tube.

For an oil flow rate of about 100 l h�1, a transition can beobserved on the drops diameters in the twisted pipe chaotic flow.In fact, at this stage, which corresponds to a Dean number of about60, the chaotic advection becomes significant. These resultsconfirm the idea that the chaotic advection flow improves theliquid/liquid dispersion.

For the same runs, the standard deviation of the drop sizedistribution, shown in Fig. 7, is about 20% less for the twisted pipethan for the helically coiled pipe. This trend appears more clearly

Measured maximum droplet diameters.

Flow rate(l h�1)

Straight pipe(mm)

Helically coiledpipe (mm)

Chaotic twistedpipe (mm)

40 1.58 1.39 1.2850 1.56 1.24 1.1160 1.20 1.12 1.0170 1.19 1.08 0.9480 0.98 0.92 0.8490 0.96 0.89 0.79

100 0.84 0.79 0.57

40 60 80 100 120 1400.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Straight Pipe

Helically coiled

Twisted pipe

Exp

erim

enta

l max

imu

m d

iam

eter

s (m

m)

Flow rate Q (l h-1)

Fig. 6. Experimental dmax versus oil flow rate.

0 20 40 60 80 100 1200.04

0.06

0.08

0.10

0.12

0.14

0.16

0.18

0.20 Helically coiled

Twisted pipe

Fitted curves:

Helically coiled

Twisted pipe

β=

sin

h( k

)

Dean number, De

Fig. 7. Droplet diameter dispersion factor for helically coiled and chaotic twisted pipe – curves fitted using Savitzky–Golay method.

C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497 489

for Dean numbers higher than 50, where the chaotic behaviourbegins to induce strains of the same magnitude as the shear flow.The standard deviation seems minimal at low Dean numberswhere the mixing process is globally homogeneous over the wholetest section but has the lowest mixing efficiency. Increasing theDean number increases the standard deviation; i.e. the non-homo-geneity of the observed emulsion is increased because of the poorradial distribution of the fluid in the mixer cross-section until aneffective Dean number is achieved at which the homogeneity andmixing efficiency are intensified.

The Sauter mean diameter d32 is a statistical parameter that canbe used to characterize the drop size distribution in the flow. Thisdiameter, given in Eq. (5), represents the mean surface diameter:

d32 ¼R‘3f ð‘Þd‘R‘2f ð‘Þd‘

ð5Þ

where f ð‘Þ is the distribution function representing the proportionof drops having a given diameter ‘ in the observed emulsion.

The proportionality between the Sauter mean diameter d32, andthe maximum drop diameter dmax is represented in Fig. 8. Theexperimental results show that the d32 and dmax diameters are lin-early correlated in the limit of validity, here 10 < De < 110. Whenthe slope of the fitted line is equal to one it implies that the Sautermean diameter is equal to the maximum drop diameter, and there-fore that the drop fragmentation is uniformly distributed over thewhole observed emulsion. The more this coefficient is close to 1,

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.00.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

1

Linear fitted curves: Helically coiled: d32 = 0.80 dmax

Chaotic twisted pipe: d32 = 0.86 dmaxExperiments:

Helically coiled Chaotic twisted pipe

Sau

ter

mea

n d

iam

eter

, d

32 (

mm

)

Maximal diameter, dmax (mm)

1

Fig. 8. Sauter mean diameter versus maximal diameter of droplets in helical coiled and chaotic advection twisted pipes.

490 C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497

the sharper is the distribution. It was found by Zhou and Kresta(1998) that the slope value for agitated tanks and bent tubesequipped with a static mixer is between 0.38 and 0.7; these valuesare much smaller than that in the present work, meaning a muchless homogeneous droplet size distribution.

To investigate the efficiency of helically coiled and chaotictwisted pipe flows, the energy cost is compared in Fig. 9 with thatof existing inline mixers reported by previous investigators (Haas,1987; Streiff et al., 1997; Lemenand et al., 2003, 2005).

The interfacial contact area A is given by the Sauter diameter:

A ¼ 6Ud32

ð6Þ

where U is the mass fraction of the dispersed phase.The energy consumption E is calculated from the pressure drop

DP:

101100

1000

10000

Energy cons

Inte

rfac

ial a

rea,

A (

m2 .m

-3)

Helically coiled pipe

Chaotic twisted pipe

Fig. 9. Comparison of the energy consumption of the helically coiled and chaoti

E ¼ DPq

ð7Þ

The pressure drop DP is obtained from the theoretical correla-tion of Ito (1969) for laminar flow in curved pipes.

From Fig. 9, the helically coiled and chaotic twisted pipe flowsare located in the small-energy consumption zone (between 1and 12 J kg�1) with a good interfacial area (between 300 and1100 m2 m�3). The Sulzer mixer seems to have the highest interfa-cial area but in the range of high energy consumption. The HEV hasthe same behavior as the geometries studied in the present work,which are both better than Kenics static mixer.

The very low energy consumption and the relatively good inter-facial contact show that helically coiled and chaotic twisted pipeflows can have good impacts in the industrial applications, espe-cially when mixing fluids that cannot tolerate high shear rates.

100

omption, E (J.kg-1)

HEV

Kenics

Sulzer

c twisted pipes with classical static mixers (data from Thakur et al., 2003).

10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101 102 10310-2

10-1

100

101

102

103

c

dpμμ=

Simple shear

2D elongation

Cacr elongation

Cacr simple shearCa

Fig. 10. Critical capillary number for shear and elongation rates; curves adapted from Grace (1982).

C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497 491

3. Theoretical prediction of drop diameters

3.1. Breakup theory for laminar flows

The Taylor theory for drop breakup is based on a balance amongthe following forces acting on a single drop of an immiscible fluidin a continuous fluid flow:

– The external viscous flow forces that tend to deform thedrop; for example, in a shear flow the viscous flow forcecan be written as:

s ¼ lc_c ð8Þ

where _c is the shear rate.– The interfacial or capillary forces that tend to preserve the

spherical shape of a drop of diameter ddrop; the pressure dif-ference at the interface balances the interfacial tension r asexpressed by the Laplace formula

Pint � Pext ¼4r

ddropð9Þ

ðPint and PextÞ are the interior and the exterior pressures,respectively.

– The internal viscous forces that resist deformation.

In order to characterize the relative effect of the different forces,a dimensionless parameter Ca (the capillary number) is defined:

Ca ¼ Viscous forcesCapillary forces

¼ ddrops2r

ð10Þ

Grace (1982) considered that drop breakup occurs when thecapillary number exceeds a critical value that depends upon theviscosity ratio (the dispersed-phase viscosity by the continuous-phase viscosity) as the manifestation of the drop’s internal viscousresistance.

The dimensional values of the maximum diameter that canwithstand an existing stress s can hence be expressed by theequation

dmax ¼2rs

Cacr ð11Þ

The critical capillary numbers are given by the Grace experi-mental curves (Fig. 10) for simple shear flows ðCacr;shearÞ and elon-gational flows ðCacr;elongationÞ. The effective capillary number inlaminar dispersion phenomena depends on the critical capillarynumber, Cacr, of both simple shear and elongation, and on the ratioof these deformation rates C ¼ _emax= _cmax.

By making simple linear interpolation we can find Cacr from thefollowing expression:

Cacr ¼ ð1� CÞCacr;shear þ CCacr;elongation ð12Þ

Knowing the viscous stresses, we obtain the C value, and bysubstituting Eqs. (8) and (12) in Eq. (11), we can predict the max-imum equilibrium diameter of the drop size distribution.

3.2. Determination of the strain rates in Dean flow

The analytical expressions for the velocity field in Dean flow in acurved channel were obtained from Dean’s asymptotic solution(Dean, 1928). In the present work, the analytical solution was cal-culated from the stream function given by Jones et al. (1989)expressed in a local frame of reference, as shown in Fig. 11. Allcomputations were carried out with MATLABTM. Table 4 presentsthe dimensionless variables involved in the analysis.

As in all the previous work, the parabolic axial velocity profile isassumed

w ¼ 2Rað1� r2Þ ð13Þ

The potential stream function in the cross-section is given by

w ¼ R

Wa2

mDe72ð4� r2Þð1� r2Þ2y ð14Þ

where r2 ¼ x2 þ y2 is the radial coordinate in the tube cross-section,m the effective kinematic viscosity, and W the flow mean velocity.

In the local coordinate system ðx; y; zÞ, the secondary velocity uand v can be written as

Table 4Flow variables.

Dimensionless variable Dimensional variable

Radial distance r r0 ¼ arTime t t0 ¼ R

W tCoordinate x x0 ¼ axCoordinate y y0 ¼ ayAxial coordinate z z0 ¼ azVelocity component u u0 ¼ a W

R uVelocity component v v 0 ¼ a W

R vAxial velocity w w0 ¼ a W

R wShear rate _c _c0 ¼ W

R_c

Elongation rate _e _e0 ¼ WR

_eStream function w ¼ R

Wa2mDe72 ð4� r2Þð1� r2Þ2y w0 ¼ Wa2

R w

Reynolds number Re ¼ 2 Wam

Dean number De ¼ a4R Re2

x

y

x y

z

θ

R

r

ax

y

x y

z

θ

R

r

a

Fig. 11. Toroidal coordinates.

492 C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497

u ¼ owoy

v ¼ � owox

ð15Þ

which leads to the following two equations:

u ¼ að1� r2Þ½6y2ð3� r2Þ � ð4� r2Þð1� r2Þ�v ¼ �6axyð1� r2Þð3� r2Þ

ð16Þ

where a ¼ Re144.

For the fully developed flow, the velocity field in the pipe cross-section appears to be independent of z. The effect of the radius ofcurvature is not explicit in this system because of the dimension-less scaling of the coordinates.

The generalized shear rate _c and elongation rate _e given by Eq.(17), and calculated from the analytical derivatives of the velocitycomponents, are given in Germain (1962) and also in Bird (2007),and are defined, respectively, as the second invariant of the defor-mation rate tensor and the extension along the axis carried by thefirst eigenvector of the deformation rate tensor. These expressionsare also used by Khakhar and Ottino (1986).

_c¼ 2 ouox

� �2þ2 ov

oy

� �2

þ2 owoz

� �2þ ou

oyþovox

� �2

þ ouozþ

owox

� �2þ ov

ozþowoy

� �2!1=2

_e¼u2 ou

ox

� �þv2 ov

oy

� �þw2 ow

oz

� �þuv ou

oyþovox

� �þuw ou

ozþowox

� �þvw ov

ozþowoy

� �u2þv2þw2

8>>>>>>>>>>><>>>>>>>>>>>:

ð17Þ

In a flow, it is the maximal value of the shear and elongation rates,_cmax helic and _emax helic, that determine the droplets’ maximum diam-eter. These values are obtained from Eq. (17) and are expressed as:

_cmax helic ¼ 4 Ra 1þ 1

150aR De

� �0:5

_emax helic ¼ 1:1 Ra 1þ 2� 10�7 a

R De3h i0:7

8<: ð18Þ

The classical dimensionless maximal shear rate for the straightpipe is given by

_cmax straight tube ¼ 4Ra

ð19Þ

For the range of Dean numbers studied here, the maximumshear and elongation rates in the flow cross-section fitted withtrend curves (Eqs. (18) and (19)) are presented in Fig. 12 for usein the dispersion model. It can be noted that, following the analyt-ical solution, the helical-pipe flow shear rate begins to deviate fromthat of the straight pipe flow at an estimated Dean number aroundDe ¼ 55. On the other hand, for a Dean number of about 600, theelongation rate begins to dominate the shear rate and governsthe dispersion process. This would lead to a decrease of 1/3 inthe diameter and would enhance the eventual benefits of the cha-otic advection flow. This range of Dean number, however, cannotbe achieved in the present experimental setup and will not be dis-cussed further.

The theoretical maximum diameters of the drops can be calcu-lated by using Eq. (11) coupled with the above maximum shearrates (Eqs. (18) and (19)). Theoretical and experimental maximumdiameters are compared in Fig. 13. The theoretical curve is uniquefor the twisted-pipe flow (chaotic advection) and the helicallycoiled pipe flow, since the Eulerian velocity fields are the same inboth cases.

The experimental maximum diameters of the drops are alsoshown in Fig. 13. At least a part of the discrepancy between exper-imental and theoretical results can be explained by the numeroushypotheses of the theoretical models, both the Taylor–Grace model(based on an assumed dynamic equilibrium situation) and theDean–Jones equations (which give an approximate solution forthe velocity field). Nevertheless, the maximum global differencewith experimental results does not exceed 10%, showing that thepresent theoretical approach captures the basic physics underlyingthe problem.

It can be noticed in Fig. 13 that the maximum drop diametersmeasured in the chaotic flow are smaller than in the helicallycoiled pipe flow, suggesting that, in addition to the deformationrates responsible for drop breakup in the helically coiled pipe flow,there exists another mechanism that leads to better drop fragmen-tation in the chaotic advection flow. It is shown in the next sectionthrough a Lagrangian analysis that this mechanism is, in fact, theway in which droplets are randomly passed to zones of highershear and strain rates in the chaotic advection flow compared tothe helically coiled flow.

4. Lagrangian analysis of the flow

4.1. Mechanical history

This Lagrangian study is based on following a fluid particle tra-jectory from its initial location at the inlet up to the outlet, as wellas on tracking the shear and elongations to which it is submitted,i.e. its ‘‘mechanical history”. The Lagrangian equations for the pas-sive advection of a fluid particle in a three-dimensional space aregiven by the following dynamical system:

_x ¼ uðx; y; z; tÞ_y ¼ vðx; y; z; tÞ_z ¼ wðx; y; z; tÞ

8><>: ð20Þ

The equations are the same for helically coiled and twisted pipeflows. For the helically coiled pipe, there is no change of the framereference axes while passing from one elbow to another. For thetwisted pipe flow, the frame reference axes are rotated of ±90� asfollows: at the end of each elbow of even number, a rotation of

101 102 103

10

100

1000

10000

Dim

ensi

on

less

str

ain

rat

es

Reynolds number, Re

Straight pipe computed shear rate Dean flowcomputed shear rate Dean flow computed extensional rate

Fig. 12. Computed strain rates in straight pipe and helically coiled pipe – maximum values in pipe cross-section.

08060402

1

2

3

Reynolds number, Re

Theoretical straight pipe

Experimental straight pipe

Max

imu

m d

iam

eter

s, d

max

(m

m)

(a)

08060402

1

2

3 Theoretical helically coiled pipe

Experimental helically coiled pipe

Experimental chaotic twisted pipe

Max

imu

m d

iam

eter

s, d

max

(m

m)

Reynolds number, Re

(b)

Fig. 13. Droplet diameters – experiments compared with theoretical values: (a)straight pipe, and (b) helically coiled and chaotic twisted pipes.

C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497 493

90� is applied to the reference axes (i.e. we replace x by y, and y by�x in the velocity equations), otherwise, at the end of each elbowof odd number, a rotation of �90� is applied.

Integration along the trajectory is carried out in the numericalprocedure by a fourth-order Runge–Kutta scheme. The passage ofa particle from the outlet of one bend to the inlet of the next mustbe calculated very accurately; the convergence is realized by aNewton–Raphson iterative process.

The sensitivity of the trajectory to the time step dt was com-puted by studying the time step at which the solution convergesto a constant value for different Dean numbers and the final posi-tion of the particle becomes independent of the maximal dt valuegiven in

dt ¼p2

Ra

� W� 10�4 ð21Þ

Results for Dean number 150 are illustrated in Fig. 14, wherethe trajectories of a passive fluid particle in the helically coiled pipeflow are compared with those of a twisted-pipe flow along 25bends and for two initial positions. In the chaotic flow, the particlesweeps all the cross-section of the pipe, while in helically coiledflow the particles remain on the same trajectory, depending ontheir initial positions. It can even be noted that in the helicallycoiled flow, for injection location x0 ¼ 0; y0 ¼ 0:43, the particleleaves the last bend at exactly the same cross-sectional positionat which it entered the pipe; this point is the centre of the Deanroll-cell at which u ¼ v ¼ 0 and therefore there is no radialvelocity.

Fig. 15 shows the Poincaré sections for the two configurations,helically coiled and chaotic, and for an initial position of a disk ofdiameter equal to 5% of the pipe cross-sectional diameter, madeup of 5000 neighbouring points. It is observed again that in thehelically coiled configuration, the particles remain in the Deanroll-cell, while in the chaotic advection case the particles spreadover the whole tube cross-section. This has a direct consequencefor the mechanical history of the fluid particle. In fact, along itschaotic trajectories the particle visits the zones of maximum shearand elongational rates. Figs. 16 and 17 show the particle’s mechan-ical history, that is, the dimensionless shear and elongation ratesthat a fluid particle undergoes at each moment along 25 bends,

Fig. 14. Trajectories of passive tracer ðDe ¼ 150; Re ¼ 81; Nc ¼ 25Þ.

Fig. 15. Poincaré sections ðDe ¼ 150; Re ¼ 81; Nc ¼ 25Þ.

494 C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497

0 5 10 15 20 25 300

2

4

6

8

10

12

14

16

18

20

22 Chaotic configuration

Helically coiled configuration

Dim

ensi

on

less

sh

ear

rate

s

Dimensionless time

Fig. 16. Mechanical history of shear rates ðx0 ¼ 0; y0 ¼ 0:75; De ¼ 100Þ.

C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497 495

for De ¼ 100 and initial position x0 ¼ 0; y0 ¼ 0:75. In the helicallycoiled configuration, the particle has a periodic trajectory andremains caged between two limiting (maximum and minimum)values of the viscous stress. On the other hand, in a chaotic flowthe particle randomly undergoes all levels of strain rates, especiallythe maximum values that are more efficient for drop breakup.

4.2. Residence time distribution (RTD)

The residence time distribution can be established by comput-ing the trajectories for a sufficiently large number of particles(about 6000), which are uniformly distributed in the inlet planesection. Fig. 18 presents the RTD as a function of dimensionlessresidence time defined in Table 4 for helically coiled pipe flow

0 5 100.0

0.2

0.4

0.6

0.8 Chaotic co

Helically co

Dim

ensi

on

less

elo

ng

atio

n r

ates

Dimensio

Fig. 17. Mechanical history of absolute elong

and twisted-pipe chaotic advection flow. In this figure are alsosuperposed the RTD for a straight pipe flow and an axial dispersedplug flow as expressed in Eqs. (22) and (23), respectively:

f ðtÞ ¼ 12

PeL

pH

� �12

exp � PeLð1�HÞ2

4H

!ð22Þ

f ðtÞ ¼ 12tH2 H t � tm

2

� �ð23Þ

where t is the residence time, tm ¼ LW the mean residence time and

H ¼ ttm

the reduced time.It can be seen from Fig. 17 that even for a small Dean number of

100, while the helically coiled pipe flow presents a RTD profile sim-ilar to that of straight pipe, the RTD profile of the chaotic advection

15 20 25 30

nfiguration

iled configuration

nless time

ation rates ðx0 ¼ 0; y0 ¼ 0:75; De ¼ 100Þ.

0 20 40 60 80 100

0.00

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.10

Helically coiled pipe RTD

Straight pipe model

Twisted pipe RTD

Axial dispersed model

Fre

qu

ency

Dimensionless residence time (s)

Fig. 18. Dimensionless residence time distribution: 25 bends for De ¼ 100.

496 C. Habchi et al. / International Journal of Multiphase Flow 35 (2009) 485–497

twisted pipe can be modeled by an axial dispersed plug flow. Inthis situation, the Péclet number based on the pipe length L,PeL ¼ WL

Dax, is about 43, where, Dax is the axial diffusion coefficient.

In agreement with the trajectories presented in Fig. 13, theresidence time dispersion is narrower in the chaotic advection con-figuration, accounting for the radial transfer enhancement. Thisresult was also found by Castelain et al. (1997) in studying theRTD in chaotic twisted pipe and helically coiled pipe flows.

5. Conclusions

Careful experiments were carried out on liquid/liquid disper-sion in oil/water flows to assess the effects of chaotic advectionon droplet breakup, in terms of both the mean diameter and thesize homogeneity. Both properties have strategic implications inmany technological processes in the pharmaceutical and cosmeticsindustries, as well as in new biological renewable-energy pro-cesses. These experiments showed that the chaotic advection flowgenerated in the twisted pipe clearly increases the efficiency ofcontactors and mixers by providing smaller and more homoge-neously dispersed droplets. The energy expenditure to obtain thisdispersion remains similar to that of the rival technology, the heli-cally coiled mixer.

To clarify the physical mechanism underlying this high disper-sion efficiency of the chaotic advection flow compared to similarlaminar flows, and also for the future design and optimization ofmultifunctional heat exchangers and reactors, a mechanistic mod-elling of this flow was undertaken. Both Eulerian and Lagrangianapproaches were applied to three flow geometries in the laminarregime: straight, helically coiled and twisted pipes. While in mostfluid dynamics problems the ultimate aim is to obtain the velocityfield in the flow, in the chaotic advection problem the velocity fieldis the starting point. To this end, we used the analytical expressionof the stream function in a curved pipe obtained by Dean (1927) asthe building block of the model. Shear and elongation rates werethen calculated and were used in conjunction with Taylor–Gracetheory to study droplet breakup.

It was shown that the Eulerian approach cannot explain thehigher dispersion efficiency of the chaotic advection flow. On the

contrary, the Lagrangian approach allows calculation of the fluidparticle trajectories and especially the ‘‘mechanical history” of afluid particle. From this we showed that chaotic advection causesfluid particles to make random visits to zones of high shear andelongation rates and therefore contributes further to dropletbreakup.

The model also allowed calculation of the RTD of fluid particlesin helically coiled and twisted-pipe flows. The RTD analysisrevealed that even for very small Dean number laminar flows,the chaotic advection flow shows a RTD distribution as narrow asan axially dispersed plug flow, while the helically coiled tube flowhas a residence time distribution similar to a straight tube flow.Thus in liquid/liquid dispersion the chaotic advection flow hastwo principal advantages over its counterpart helically coiled pipeflow: first, generation of smaller droplets, second, a more homog-enous droplet diameter distribution.

The mechanistic model developed in this work provides a solidbasis for obtaining physical insight into dispersion phenomena bylaminar flows and offers a powerful design and optimization toolto designers of future innovative devices. Future work will focuson gas/liquid dispersion by chaotic advection.

Acknowledgements

The authors gratefully acknowledge Dr. P. Carrière of LMFA(Laboratoire de Mécanique des Fluides et d’Acoustique) of ÉcoleCentrale de Lyon (France), for fruitful and enlighteningdiscussions.

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